The ratio of ages of A and B 4 years ago was 5:3. The sum of present ages of A, B and C is 80 years. If present age of C is equal to sum of present ages of A and B.find the present age of A.

Options

A. A)17 years
B. B)24 years
C. C)20 years
D. D)22 years

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Set up the equations based on the given information.
Let the present ages of A and B be \( a \) and \( b \) respectively. According to the problem, the ratio of their ages 4 years ago was 5:3. This can be expressed as:
\[ \frac{a - 4}{b - 4} = \frac{5}{3} \]
Cross-multiplying gives us:
\[ 3(a - 4) = 5(b - 4) \]
Expanding this, we get:
\[ 3a - 12 = 5b - 20 \]
Rearranging the equation leads to:
\[ 3a - 5b = -8 \quad \text{(Equation 1)} \]

### Step 2: Use the information about the sum of ages.
We know that the sum of the present ages of A, B, and C is 80 years:
\[ a + b + c = 80 \]
We are also told that the present age of C is equal to the sum of the present ages of A and B:
\[ c = a + b \]
Substituting this into the first equation gives:
\[ a + b + (a + b) = 80 \]
This simplifies to:
\[ 2(a + b) = 80 \]
Dividing both sides by 2 results in:
\[ a + b = 40 \quad \text{(Equation 2)} \]

### Step 3: Substitute Equation 2 into Equation 1.
From Equation 2, we can express \( b \) in terms of \( a \):
\[ b = 40 - a \]
Now, substitute this expression for \( b \) into Equation 1:
\[ 3a - 5(40 - a) = -8 \]
Expanding this gives:
\[ 3a - 200 + 5a = -8 \]
Combining like terms results in:
\[ 8a - 200 = -8 \]
Adding 200 to both sides gives:
\[ 8a = 192 \]
Dividing both sides by 8 results in:
\[ a = 24 \]

### Step 4: Find the present age of B.
Using the value of \( a \) in Equation 2:
\[ b = 40 - 24 = 16 \]

### Step 5: Verify the solution.
Now we can verify:
- The ages of A and B 4 years ago:
- A's age: \( 24 - 4 = 20 \)
- B's age: \( 16 - 4 = 12 \)
- The ratio \( \frac{20}{12} = \frac{5}{3} \) confirms the initial condition.
- The sum of their present ages plus C: \( 24 + 16 + (24 + 16) = 80 \) also holds true.

Thus, the present age of A is:
\[ \boxed{24} \]